Gravitating fluids with Lie symmetries

TL;DR

This paper explores a nonlinear PDE related to gravitating fluid plates, using Lie symmetries to find new explicit solutions and classify existing ones, including solutions with a linear barotropic equation of state.

Contribution

It applies Lie symmetry analysis to reduce and solve a complex PDE in gravitating fluids, discovering new solution families and characterizing known solutions.

Findings

Found new explicit solutions to the PDE.

Characterized solutions with a linear barotropic equation of state.

Identified the ODEs for all one-dimensional optimal subgroups.

Abstract

We analyse the underlying nonlinear partial differential equation which arises in the study of gravitating flat fluid plates of embedding class one. Our interest in this equation lies in discussing new solutions that can be found by means of Lie point symmetries. The method utilised reduces the partial differential equation to an ordinary differential equation according to the Lie symmetry admitted. We show that a class of solutions found previously can be characterised by a particular Lie generator. Several new families of solutions are found explicitly. In particular we find the relevant ordinary differential equation for all one-dimensional optimal subgroups; in several cases the ordinary differential equation can be solved in general. We are in a position to characterise particular solutions with a linear barotropic equation of state.